Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions

Abstract

Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output. However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians. We introduce an alternative computational "inverse method," the eigenstate-to-Hamiltonian construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems. EHC takes as input a wave function jψ T i and produces as output Hamiltonians for which jψ T i is an eigenstate. This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix. EHC is widely applicable to a number of models and, in this work, we consider seven different examples. Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model. EHC gives an alternative inverse approach for studying quantum many-body phenomena.

Learning the dynamics of open quantum systems from their steady states

“…An isolated quantum system can be characterized by learning its underlying Hamiltonian. This can be achieved by monitoring the dynamics that the Hamiltonian generates [18][19][20][21][22][23][24][25][26][27][28][29][30][31], or by measuring local observables in one of its eigenstates or thermal states [32][33][34][35][36][37][38][39][40]. However, realistic quantum systems are never fully isolated.…”

Automatic design of Hamiltonians

“…One of these tasks, finding the generating (entanglement) Hamiltonian given an eigenstate attracted interest in literature recently [7,18,41,56,62]. Given a reference wavefunction one chooses a set of hermitian operatorsĥ i that form an operator basis in the considered Hilbert space or its subspace such that the Hamiltonian of interest can be expanded as H = i α iĥi .…”

Parent Hamiltonian reconstruction of Jastrow-Gutzwiller wavefunctions

“…Very recently, a series of novel techniques based on systematic approaches have been considered in Ref. [41][42][43][44][45]. Indeed, the authors of the latter works introduced new efficient computational algorithms, which remarkably scale polynomially in the system size when restricting the search to local Hamiltonians that have a given initial state as the input eigenstate.…”